Question

Give a careful statement of Euler's Formula; solve all z^8=1, sketch a plot for z^8=1, and use your plot to explain why the solution adds up to 0 (if roots have multiplicity, add them repeatedly according to their multiplicity).

Answer 1

Euler's Formula is e^ix = cos(x) + i · sin(x), and solving z^8 = 1 yields the eight roots of unity on the complex plane unit circle. When plotted, these roots display symmetry around the origin, resulting in their sum being 0 as they cancel each other out.

Step-by-step explanation:

Euler's Formula states that for any real number x, eix = cos(x) + i · sin(x), where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the cosine and sine functions, respectively.

To solve z8 = 1, we realize 1 can be written as e0i, which by Euler's formula is also cos(0) + i · sin(0). The solutions to z8 = 1 will be the 8th roots of unity, which are equidistant points on the unit circle in the complex plane. These sixt different angles of 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, and 7π/4 radians correspond to the complex numbers 1, i, -1, -i, 1, i, -1 and -i. These eight complex numbers are the solutions, and when added up, each number is canceled out by its opposite, summing up to 0.

When we sketch a plot for z8 = 1, it will show the eight points on the unit circle in the complex plane spread evenly at π/4 radians apart. The symmetrical nature of these solutions about the origin explains why the sum of the solutions of z8 = 1 is 0, as each root is paired with its negative counterpart, canceling each other out.